What is the difference between a discrete probability...

Key Concepts

Discrete Probability Function

A discrete probability function, also known as a probability mass function (PMF), is used with discrete random variables. It assigns a specific probability to each possible outcome, and the sum of all these probabilities equals 1. This function deals with scenarios where the set of possible outcomes is countable, such as the roll of a die or the number of emails received in a day.

Probability Density Function

A probability density function (PDF) is used with continuous random variables. Unlike a PMF, a PDF does not give probabilities directly, but instead provides a density value; the probability that the random variable falls within a certain range is determined by integrating the PDF over that range. The total area under the curve of a PDF, across its entire domain, equals 1.

Discrete vs. Continuous Random Variables

The key distinction between discrete and continuous random variables underlies the difference between a discrete probability function and a probability density function. Discrete random variables take on countable values and are associated with PMFs, making it possible to assign exact probabilities to each outcome. Continuous random variables, on the other hand, take on an uncountable number of values, which necessitates the use of PDFs and integration to calculate probabilities over intervals.

Summation vs. Integration

In the context of probability functions, discrete probability functions utilize summation to obtain probabilities since the outcomes are countable, with the sum of all probabilities being equal to 1. In contrast, probability density functions require integration over an interval to determine the probability that a continuous random variable falls within that interval. This distinction in mathematical treatment reflects the inherent differences between discrete and continuous probability distributions.

Transcript

00:01 Okay, so we want to talk about the discrete probability function and the probability density function and the difference between the two of them.

00:08 So in the discrete probability function, we have a finite number of outcomes, right? this could be a question like, what dice roll do we expect or are we going to get heads or tails, right? there's two sides of the coin, there's six sides of the dice, stuff like this where we have a finite number.

00:30 Of outcomes.

00:33 Finite number of outcomes.

00:40 On the other hand, with the probability density function, we have an infinite number of outcomes, infinite number of outcomes, because we're going to be defining this function on the real number line.

00:55 So even if our probability density function is only defined between, say, one and two, there's still an infinite number of possibilities between one and two that we can pick.

01:07 Examples of this might be like cost of something or height if you measure it very precisely and you let any any value say between 60 inches and 70 inches be a possible value.

01:26 We can always zoom in closer and closer on the interval and find other values that we can be right.

01:34 So the next main difference between the two of these is is in how they're graphed.

01:39 So the discrete probability function, if we wanted to describe that, since we have only a finite number of outcomes, let's say this is maybe like a coin flip example.

01:55 We just have heads and tails.

01:58 And over here we'd have our y -axis and a value of 0 .5...

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